| L(s) = 1 | − 4·7-s − 2·13-s + 6·17-s + 4·19-s + 6·29-s − 8·31-s − 2·37-s + 6·41-s − 4·43-s + 9·49-s − 6·53-s − 10·61-s − 4·67-s − 2·73-s − 8·79-s − 12·83-s − 18·89-s + 8·91-s − 2·97-s − 18·101-s − 4·103-s + 12·107-s − 10·109-s − 18·113-s − 24·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.824·53-s − 1.28·61-s − 0.488·67-s − 0.234·73-s − 0.900·79-s − 1.31·83-s − 1.90·89-s + 0.838·91-s − 0.203·97-s − 1.79·101-s − 0.394·103-s + 1.16·107-s − 0.957·109-s − 1.69·113-s − 2.20·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.101485244373939343160290559425, −7.35042654382863533673550380313, −6.78354731906038205235867209347, −5.87146842228725166509134997887, −5.34762325653913158633333815914, −4.22720411424127134624171462120, −3.25067910161980170709704604527, −2.84728890251355077691674063068, −1.34557356174820418480074519421, 0,
1.34557356174820418480074519421, 2.84728890251355077691674063068, 3.25067910161980170709704604527, 4.22720411424127134624171462120, 5.34762325653913158633333815914, 5.87146842228725166509134997887, 6.78354731906038205235867209347, 7.35042654382863533673550380313, 8.101485244373939343160290559425
